Three-dimensional Linear System Research Articles

Li-Yorke chaos in weak topology of the n-dimensional linear systems

The paper deals with the complicated dynamics of the linear systems on the n-dimensional Euclidean space with the weak topology. A weak topology defined by a family of semi-norms is found to systematically investigate the chaotic dynamics in weak topology. It is rigorously proved that the one-dimensional linear systems are not Li-Yorke chaotic in weak topology and the n-dimensional linear systems with n>1 exhibit weak Li-Yorke chaos in weak topology. Moreover, some necessary and sufficient conditions for weak Li-Yorke chaos in weak topology of n-dimensional linear systems with n≥2 are established by proving the existence of a semi-irregular or an irregular vector in weak topology. As an application, the dynamics in weak topology for the three-dimensional linear systems with real distinct, real repeated and complex conjugate eigenvalues are classified.

Hopf bifurcation at infinity in 3D Relay systems

A complete analysis of the limit cycle bifurcation from infinity in 3D Relay systems, which belong to the class of three-dimensional symmetric discontinuous piecewise linear systems with two zones, is presented. A criticality parameter is found, whose sign determines the character of the bifurcation. When such non-degeneracy parameter vanishes, a higher co-dimension bifurcation takes place, giving rise to the emergence of a curve of saddle–node bifurcations of periodic orbits, which allows to determine parameter regions where two limit cycles coexist.The existence of a large amplitude limit cycle that bifurcates from infinity is justified through a suitable adaptation of the closing equations method, and analytical expressions for its amplitude and period are provided. Derivatives of the corresponding transition maps are rigorously studied in order to characterize the stability of the bifurcating limit cycle.The theoretical results are applied to a specific family of 3D relay systems, where several high co-dimension bifurcation points are detected, organizing the bifurcation set of the family.

Initial-offset boosted dynamics in memristor-sine-modulation-based system and its image encryption application

The initial-offset boosted dynamics has attracted much attentions for its potential applications in synchronous control and secure communication. This paper builds a five-dimensional (5-D) memristor-sine-modulation-based (MSMB) system to generate initial-offset boosted dynamics. The 5-D MSMB system is built by adding a negative feedback loop and one positive feedback loop to a three-dimensional linear system, that is, both the memristor- and sine-modulation methods are used. Through theoretical analysis, the plane equilibrium set with its initial-related stability and the periodical evolutionary dynamics together with its mechanism are revealed. Furthermore, initial-offset boosted dynamics induced by the memristor initial conditions (MICs) and the non-memristor initial conditions (NMICs) is uncovered by numerical simulations. Several sets of coexisting initial-offset boosted attractors are also captured by Microcontroller-based digital implementation, which are well matched with the numerical results. Finally, three sets of the pseudo random sequences generated from the MSMB system are applied to image encryption with good encryption performance.

Pseudo-transient ghost fluid methods for the Poisson-Boltzmann equation with a two-component regularization

The Poisson Boltzmann equation (PBE) is a well-established implicit solvent continuum model for the electrostatic analysis of solvated biomolecules. The numerical solution of the nonlinear PBE is still a challenge due to its exponential nonlinear term, strong singularity by the source terms, and distinct dielectric regions. In this paper, a new pseudo-transient approach is proposed, which combines an analytical treatment of singular charges in a two-component regularization, with an analytical integration of nonlinear term in pseudo-time solution. To ensure efficiency, both fully implicit alternating direction implicit (ADI) and unconditionally stable locally one-dimensional (LOD) methods have been constructed to decompose three-dimensional linear systems into one-dimensional (1D) ones in each pseudo-time step. Moreover, to accommodate the nonzero function and flux jumps across the dielectric interface, a modified ghost fluid method (GFM) has been introduced as a first order accurate sharp interface method in 1D style, which minimizes the information needed for the molecular surface. The 1D finite-difference matrix generated by the GFM is symmetric and diagonally dominant, so that the stability of ADI and LOD methods is boosted. The proposed pseudo-transient GFM schemes have been numerically validated by calculating solvation free energy, binding energy, and salt effect of various proteins. It has been found that with the augmentation of regularization and GFM interface treatment, the ADI method not only enhances the accuracy dramatically, but also improves the stability significantly. By using a large time increment, an efficient protein simulation can be realized in steady-state solutions. Therefore, the proposed GFM-ADI and GFM-LOD methods provide accurate, stable, and efficient tools for biomolecular simulations.

Hopf bifurcation at infinity in 3D symmetric piecewise linear systems. Application to a Bonhoeffer–van der Pol oscillator

In this work, a Hopf bifurcation at infinity in three-dimensional symmetric continuous piecewise linear systems with three zones is analyzed. By adapting the so-called closing equations method, which constitutes a suitable technique to detect limit cycles bifurcation in piecewise linear systems, we give for the first time a complete characterization of the existence and stability of the limit cycle of large amplitude that bifurcates from the point at infinity. Analytical expressions for the period and amplitude of the bifurcating limit cycles are obtained. As an application of these results, we study the appearance of a large amplitude limit cycle in a Bonhoeffer–van der Pol oscillator.

Memristor initial-boosted coexisting plane bifurcations and its extreme multi-stability reconstitution in two-memristor-based dynamical system

Initial-dependent extreme multi-stability and offset-boosted coexisting attractors have been significantly concerned recently. This paper constructs a novel five-dimensional (5-D) two-memristor-based dynamical system by introducing two memristors with cosine memductance into a three-dimensional (3-D) linear autonomous dissipative system. Through theoretical analyses and numerical plots, the memristor initial-boosted coexisting plane bifurcations are found and the memristor initial-dependent extreme multi-stability is revealed in such a two-memristor-based dynamical system with plane equilibrium. Furthermore, a dimensionality reduction model with the determined equilibrium is established via an integral transformation method, upon which the memristor initial-dependent extreme multi-stability is reconstituted theoretically and expounded numerically. Finally, physically circuit-implemented PSIM (power simulation) simulations are carried out to validate the plane offset-boosted coexisting behaviors.

A Lorenz-type attractor in a piecewise-smooth system: Rigorous results.

Chaotic attractors appear in various physical and biological models; however, rigorous proofs of their existence and bifurcations are rare. In this paper, we construct a simple piecewise-smooth model which switches between three three-dimensional linear systems that yield a singular hyperbolic attractor whose structure and bifurcations are similar to those of the celebrated Lorenz attractor. Due to integrability of the linear systems composing the model, we derive a Poincaré return map to rigorously prove the existence of the Lorenz-type attractor and explicitly characterize bifurcations that lead to its birth, structural changes, and disappearance. In particular, we analytically calculate a bifurcation curve explicit in the model's parameters that corresponds to the formation of homoclinic orbits of a saddle, often referred to as a "homoclinic butterfly." We explicitly indicate the system's parameters that yield a bifurcation of two heteroclinic orbits connecting the saddle fixed point and two symmetrical saddle periodic orbits that gives birth to the chaotic attractor as in the Lorenz system. These analytical tasks are out of reach for the original nonintegrable Lorenz system. Our approach to designing piecewise-smooth dynamical systems with a predefined chaotic attractor and exact solutions may open the door to the synthesis and rigorous analysis of hyperbolic attractors.

Stabilization of linear control systems. Pole assignment problem. A survey

This paper reviews the problems of stabilization and pole assignment in the control of linear time-invariant systems using the static state and output feedback. The main results are presented, along with a literature review. Stationary and nonstationary stabilizations with the static feedback are considered. The algorithms of low- and high-frequency stabilization of linear systems for solving Brockett’s stabilization problem are provided. The effective necessary and sufficient conditions for stabilization of two- and three-dimensional controllable linear systems are given in terms of the system parameters. The pole assignment problem and the related issues for linear systems are considered.

A novel multi-scroll chaotic generator: Analysis, simulation, and implementation**Project supported by the National Natural Science Foundation of China (Grant Nos. 51377124 and 51521065), the Foundation for the Author of National Excellent Doctoral Dissertation, China (Grant No. 201337), and the New Star of Youth Science and Technology of Shaanxi Province, China (Grant No. 2016KJXX-40).

Based on a new three-dimensional autonomous linear system and designing a specific form of saturated function series and a sign function with two variables of system, which are employed to increase saddle-focus equilibrium points with index 2, a novel multi-scroll chaotic system is proposed and its typical dynamical characteristics including bifurcation diagram, Poincare map, and the stability of equilibrium points are analyzed. The hardware circuit is designed and the experimental results are presented for confirmation.

Utilizing null controllable regions to stabilize input-constrained nonlinear systems

In this paper, we present a method for control of input-constrained nonlinear systems that offers guaranteed stabilization from the entire null controllable region (NCR). The controller achieves stabilization by using a constrained control Lyapunov function (CCLF) based on this NCR. Prior to online implementation, the level sets the CCLF are constructed using an iterative algorithm. The algorithm works by using an invariance principle to expand an initial quadratic Lyapunov function-based region of attraction. The level sets of this CCLF are then utilized in the control calculations, and in particular, an MPC is formulated that requires the system to go to lower level sets of the CCLF. The proposed MPC thus achieves stabilization from the entire NCR. The proposed approach is first corroborated against existing results for linear systems using two- and three-dimensional linear systems examples. Subsequently, the implementation is shown for two and three dimensional nonlinear systems.

Chaotification of a class of linear switching systems based on a Shilnikov criterion

This paper proposes an approach for constructing and generating chaos from a class of three-dimensional linear switching systems via a heteroclinic loop based on the Shilnikov criterion. First, the existence of a switching rule for the system is derived by utilizing the Shilnikov heteroclinic criterion. Then a general design philosophy and its procedure of switching rule are provided to ensure that the proposed approach is applicable to engineering. Two numerical examples are presented to validate the main principle and the implementability of the scheme. Theoretical analysis and numerical simulation are used to demonstrate the feasibility and effectiveness of developed techniques.

Exact Finite Difference Schemes for Three-Dimensional Linear Systems with Constant Coefficients

In this paper implicit and explicit exact difference schemes (EDS) for system $\textbf{x}' = A\textbf{x}$ of three linear differential equations with constant coefficients are constructed. Numerical simulations for stiff problem and for problems with periodic solutions on very large time interval demonstrate the efficiency and exactness of the EDS compared with high-order numerical methods. This result can be extended for constructing EDS for general systems of $n$ linear differential equations with constant coefficients and nonstandard finite difference (NSFD) schemes preserving stability properties for quasi-linear system of equations $\textbf{x}' = A\textbf{x }+ f(\textbf{x})$.

Generating Chaos from Two Three-Dimensional Rigorous Linear Systems via a Novel Switching Control Approach

By constructing two three-dimensional (3D) rigorous linear systems, a novel switching control approach for generating chaos from two linear systems is presented. Two 3D linear systems without any constant term have only one common equilibrium point that is the origin. By employing an absolute-value switching law, chaos can be generated by switching between two linear systems. Basic dynamical behaviors of the systems are investigated in detail. Numerical examples illustrate the effectiveness of the presented approach.

Delayed feedback stabilization of unstable equilibria

Analytical results of stabilizing of unstable equilibria of two- and three-dimensional linear systems are presented. The stabilization method is based on time-delayed feedback. Effective necessary and/or sufficient conditions for stabilizing the considered systems in terms of its parameters are obtained. The potential of delayed state and output feedback controls is shown. The theorems proved show that such a delayed feedback approach allows one to extend the possibilities available with static time-invariant state and output feedbacks for stabilizability of unstable linear time-invariant controllable systems. It is used two types of time-delayed feedback: conventional and by Pyragas' ones. A comparison of the stabilizability domains via both types of feedback is given.

Oblique reconstructions in tomosynthesis. I. Linear systems theory

By convention, slices in a tomosynthesis reconstruction are created on planes parallel to the detector. It has not yet been demonstrated that slices can be generated along oblique directions through the same volume, analogous to multiplanar reconstructions in computed tomography (CT). The purpose of this work is to give a proof-of-principle justification for oblique reconstructions in tomosynthesis, which acquires projection images over a smaller angular range than CT. To investigate the visibility of individual frequencies in an oblique reconstruction, a theoretical framework is developed in which the reconstruction of a sinusoidal input is calculated. The test frequency is pitched at an angle in a 2D parallel-beam acquisition geometry. Reconstructions are evaluated along the pitch of the object. The modulation transfer function (MTF) is calculated from the relative signal at various test frequencies. The MTF determines whether modulation is within detectable limits in oblique reconstructions. In the previous linear systems (LS) model [B. Zhao and W. Zhao, "Three-dimensional linear system analysis for breast tomosynthesis," Med. Phys. 35(12), 5219-5232 (2008)], the MTF was calculated only in reconstructed slices parallel to the detector. This paper generalizes the MTF calculation to reconstructed slices at all possible pitches. Unlike the previous LS model, this paper also analyzes the effect of object thickness on the MTF. A second test object that is considered is a rod whose long axis is pitched similar to the sinusoidal input. The rod is used to assess whether the length of an object can be correctly estimated in oblique reconstructions. To simulate the conventional display of the reconstruction, slices are first created along a 0° pitch. This direction is perpendicular to the rays of the central projection. The authors show that the input frequency of a pitched sinusoidal object cannot be determined using these slices. By changing the pitch of the slice to match the object, it is shown that the input frequency is properly resolved. To prove that modulation is preserved in pitched slices, the MTF is also calculated. Modulation is within detectable limits over a broad range of pitches if the object is thin, but is detectable over a narrower range of pitches if the object is thick. Turning next to the second test object, it is shown that the length of a pitched rod can be correctly determined in oblique reconstructions. Concordant with the behavior of the MTF, the length estimate is accurate over a broad range of pitches if the object is thin, but is correct over a narrower range of pitches if the object is thick. This work justifies the feasibility of oblique reconstructions in tomosynthesis. It is demonstrated that pitched test objects are most easily visualized with oblique reconstructions instead of conventional reconstructions. In order to achieve high image quality over a broad range of pitches, the object must be thin. By analyzing the effect of reconstruction pitch and object thickness on image quality, this paper generalizes the previous LS model for tomosynthesis.

Explicit Hamiltonian realizations of non-degenerate three-dimensional real linear differential systems

The purpose of this work is to give explicit Hamiltonian realizations for all non-degenerate real three-dimensional linear differential systems.

Decomposition and exact solutions of three-dimensional nonstationary linearized equations for a viscous fluid

A new exact method for solving three-dimensional linear systems of hydrodynamic equations is described based on decomposing these systems into three simpler equations. It is shown that the general solution to three-dimensional Stokes equations (when there are no mass forces) can be expressed by means of solutions to two independent equations: the heat conduction equation and the Laplace equation. A class of solutions with the linear dependence of velocity components on two space variables is studied, and their physical interpretation is given. Axial flows are considered, and certain hydrodynamic problems are solved. A general solution to three-dimensional Oseen equations is constructed. The linearized equations of motion for a viscoelastic Oldroyd fluid are studied

Unfolding the fold-Hopf bifurcation in piecewise linear continuous differential systems with symmetry

Three-dimensional symmetric piecewise linear differential systems near the conditions corresponding to the fold-Hopf bifurcation for smooth systems are considered. By introducing one small parameter, we study the bifurcation of limit cycles in passing through its critical value, when the three eigenvalues of the linear part at the origin are at the imaginary axis of the complex plane.The simultaneous bifurcation of three limit cycles is proved. Conditions for stability of these limit cycles are provided, and analytical expressions for their period and amplitude are obtained.Finally, we apply the achieved theoretical results to a generalized version of Chua’s circuit, showing that the fold-Hopf bifurcation takes place for a certain range of parameters.

KNOTTED PERIODIC SOLUTIONS OF A LINEAR NONAUTONOMOUS SYSTEM AND SOME RELATED THREE-DIMENSIONAL FLOWS

This paper considers a three-dimensional linear nonautonomous systems. It shows that for every integer frequency parameter value, this system has a distinct type of knotted periodic solutions, which lie in a bounded region of R3. Exact explicit parametric representations of the knotted periodic solutions are given. By using these parametric representations, two series of three-dimensional flows are constructed, such that these three-dimensional autonomous systems have knotted periodic orbits in the three-dimensional phase space.

Saddle–node bifurcation of invariant cones in 3D piecewise linear systems

In this work, the existence of a saddle–node bifurcation of invariant cones in three-dimensional continuous homogeneous piecewise linear systems is considered. First, we prove that invariant cones for this class of systems correspond one-to-one to periodic orbits of a continuous piecewise cubic system defined on the unit sphere. Second, let us give the conditions for which the sphere is foliated by a continuum of periodic orbits. The principal idea is looking for the periodic orbits of the continuum that persist when this situation is perturbed. To do this, we establish the relationship between the invariant cones of the three-dimensional system and the periodic orbits of two planar hybrid piecewise linear systems. Next, we define two functions whose zeros provide the invariant cones that persist under the perturbation. These functions will be called Melnikov functions and their properties allow us to state some results about the existence of invariant cones and other results about the existence of saddle–node bifurcations of invariant cones, which is the principal goal of this paper.